Saturday, December 10, 2016

Quasicrystals1,2,
short for quasiperiodic crystals, are solids able to violate the
conventional rules of crystallography because their structure is
“quasiperiodic” rather than periodic; that is, their atomic density can
be described by a finite sum of periodic functions with periods whose
ratio is irrational. Their diffraction pattern consists of true Bragg
peaks whose positions can be expressed as integer linear combinations of
D integer linearly independent wavevectors where D is greater than the
number of space dimensions. Among the quasicrystals made in the
laboratory, many exhibit a crystallographically forbidden,
three-dimensional icosahedral symmetry defined by D = 6 integer linearly
independent wavevectors. See: Collisions in outer space produced an icosahedral phase in the Khatyrka meteorite never observed previously in the laboratory

Sunday, November 29, 2015

Against symmetry, is what constitutes time as a measure. So there is this argument in there too.:)

My aim in this essay is to propose a conception of mathematics that is fully consonant with naturalism. By that I mean the hypothesis that everything that exists is part of the natural world, which makes up a unitary whole. This is in contradiction with the Platonic view of mathematics held by many physicists and mathematicians according to which,
mathematical truths are facts about mathematical objects which exist in a separate, timeless realm of reality, which exists apart from and in addition to physical reality. -A naturalist account of the limited, and hence reasonable, effectiveness of mathematics in physics

The point I think I am making, is that in issuance of any position, any
idea has to emerge from an a prior state in order for the "unitary
whole" to be fully understood? Timeless, becomes an illogical position,
since any idea in itself becomes an "asymmetrical view" as a product of
the phenomenal world. Symmetry then implies, a need for, and a better
description of the unitary whole.

There is a constant theme that I observed with Lee Smolin regarding the effectiveness of the idea about what the Platonic world means in face of being a realist of the natural world. So in one stroke, if we could but eliminate the question about the Platonic world of forms, would we see that Platonism is a duelist of nature, and not a realist of the kind that exists as a product of the natural world. But more then this, the idea somehow that the platonic world is a timeless truth about our existence.

The
great mathematician fully, almost ruthlessly, exploits the domain of
permissible reasoning and skirts the impermissible. That
recklessness does not lead him into a morass of contradictions is a
miracle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection,
to the perfection which it seems to possess. However, this is not our
present subject. The principal point which will have to be recalled
later is that the mathematician could formulate only a handful of
interesting theorems without defining concepts beyond those contained in
the axioms and that the concepts outside those contained in the axioms
are defined with a view of permitting ingenious logical operations which
appeal to our aesthetic sense both as operations and also in their
results of great generality and simplicity.

[3 M. Polanyi, in his Personal Knowledge (Chicago: University of ChicagoPress, 1958), says: "All these difficulties are but consequences of ourrefusal to see that mathematics cannot be defined without acknowledgingits most obvious feature: namely, that it is interesting" (p 188).]

So you can see that I attain one end of the argument, against being a naturalist, given I hold to views about the Platonic world? Against FXQi, and its awarding program regarding the selection of the subject as an awardee, if I counter Lee's perspective?

There are many other classes of things that are evoked. There are forms of poetry
and music that have rigid rules which define vast or countably infinite sets of possible
realizations. They were invented, it is absurd to think that haiku or the blues existed
before particular people made the first one. Once defined there are many discoveries
to be made exploring the landscape of possible realizations of the rules. A master may
experience the senses of discovery, beauty and wonder, but these are not arguments for
the prior or timeless existence of the art form independent of human creativity. See: A naturalist account of the limited, and hence reasonable,
effectiveness of mathematics in physicsBy Lee Smolin

I have my own views about what constitutes what a naturalist is in face of what Lee Smolin grants it to be in face of the argument regarding what is an false as an argument about what is invented or discovered. So of course, full and foremost, what is a naturalist?

But again, let us be reminded of the poet or the artist,

Mathematics, rightly viewed, possesses not only truth, but supreme
beauty, a beauty cold and austere, like that of sculpture, without appeal
to any part of our weaker nature, without the gorgeous trappings of
painting or music, yet sublimely pure, and capable of a stern perfection
such as only the greatest art can show. The true spirit of delight, the
exaltation, the sense of being more than Man, which is the touchstone of
the highest excellence, is to be found in mathematics as surely as in
poetry. --BERTRAND RUSSELL, Study of Mathematics

Wednesday, May 20, 2015

Kant, however, is correct in that we inevitably try and conceive of transcendent, which means unconditioned, objects. This generates "dialectical illusion" in the Antinomies
of reason. Kant thought that some Antinomies could be resolved as
"postulates of practical reason" (God, freedom, and immortality); but
the arguments for the postulates are not very strong (except for
freedom), and discarding them helps guard against the temptation of
critics to interpret Kant in terms of a kind of Cartesian
"transcendental realism" (i.e. real objects are "out there," but it is
not clear how or that we know them). If phenomenal objects, as
individuals, are real, then the abstract structure (fallibly) conceived
by us within them is also real. Empirical realism for phenomenal
objects means that an initial Kantian Conceputalism turn into a Realism for universals. See: Meaning and the Problem of Universals, A Kant-Friesian Approach

It s always interesting for me to see what constitutes a Platonist in the world today. So I had to look at this question. There always seems to be help when you need it most, so information in the truest sense, is never lacking, but readily available as if taken from some construct we create of the transcendent.

***

Platonism,
rendered as a proper noun, is the philosophy of Plato or the name of
other philosophical systems considered closely derived from it. In
narrower usage, platonism, rendered as a common noun (with a lower case
'p', subject to sentence case), refers to the philosophy that affirms
the existence of abstract objects, which are asserted to "exist" in a
"third realm" distinct both from the sensible external world and from
the internal world of consciousness, and is the opposite of nominalism
(with a lower case "n").[1] Lower case "platonists" need not accept any
of the doctrines of Plato.[1]

In a narrower sense, the term might indicate the doctrine of Platonic
realism. The central concept of Platonism, a distinction essential to
the Theory of Forms, is the distinction between the reality which is
perceptible but unintelligible, and the reality which is imperceptible
but intelligible. The forms are typically described in dialogues such as
the Phaedo, Symposium and Republic as transcendent, perfect archetypes,
of which objects in the everyday world are imperfect copies. In the
Republic the highest form is identified as the Form of the Good, the
source of all other forms, which could be known by reason. In the
Sophist, a later work, the forms being, sameness and difference are
listed among the primordial "Great Kinds". In the 3rd century BC,
Arcesilaus adopted skepticism, which became a central tenet of the
school until 90 BC when Antiochus added Stoic elements, rejected
skepticism, and began a period known as Middle Platonism. In the 3rd
century AD, Plotinus added mystical elements, establishing Neoplatonism,
in which the summit of existence was the One or the Good, the source of
all things; in virtue and meditation the soul had the power to elevate
itself to attain union with the One. Platonism had a profound effect on
Western thought, and many Platonic notions were adopted by the Christian
church which understood Plato's forms as God's thoughts, while
Neoplatonism became a major influence on Christian mysticism, in the
West through St Augustine, Doctor of the Catholic Church whose Christian
writings were heavily influenced by Plotinus' Enneads,[2] and in turn
were foundations for the whole of Western Christian thought" Platonism

***

Now beauty, as we said, shone bright among those visions, and in this
world below we apprehend it through the clearest of our senses, clear
and resplendent. For sight is the keenest of the physical senses, though
wisdom is not seen by it -- how passionate would be our desire for it,
if such a clear image of wisdom were granted as would come through sight
-- and the same is true of the other beloved objects; but beauty alone has this privilege, to be most clearly seen and most lovely of them all.
[Phaedrus, 250D, after R. Hackford, Plato's Phaedrus, Library of the
Liberal Arts, 1952, p. 93, and the Loeb Classical Library, Euthryphro
Apology Crito Phaedo Phaedrus, Harvard University Press, 1914-1966,
p.485, boldface added]

For example, thought cannot be attributed to the One because thought
implies distinction between a thinker and an object of thought (again
dyad). Even the self-contemplating intelligence (the noesis of the nous)
must contain duality. "Once you have uttered 'The Good,' add no
further thought: by any addition, and in proportion to that addition,
you introduce a deficiency." [III.8.10] Plotinus denies sentience,
self-awareness or any other action (ergon) to the One [V.6.6]. Rather,
if we insist on describing it further, we must call the One a sheer Dynamis or potentiality without which nothing could exist. [III.8.10]
As Plotinus explains in both places and elsewhere [e.g. V.6.3], it is
impossible for the One to be Being or a self-aware Creator God. At
[V.6.4], Plotinus compared the One to "light", the Divine Nous (first
will towards Good) to the "Sun", and lastly the Soul to the "Moon"
whose light is merely a "derivative conglomeration of light from the
'Sun'". The first light could exist without any celestial body. Plotinus -

***

"...underwriting the form languages of ever more domains of
mathematics is a set of deep patterns which not only offer access to a
kind of ideality that Plato claimed to see the universe as created with
in the Timaeus; more than this, the realm of Platonic forms is itself
subsumed in this new set of design elements-- and their most general
instances are not the regular solids, but crystallographic reflection
groups. You know, those things the non-professionals call . . .
kaleidoscopes! * (In the next exciting episode, we'll see how Derrida
claims mathematics is the key to freeing us from 'logocentrism'-- then
ask him why, then, he jettisoned the deepest structures of mathematical
patterning just to make his name...)

* H. S. M. Coxeter, Regular Polytopes (New York: Dover, 1973) is the
great classic text by a great creative force in this beautiful area of
geometry (A polytope is an n-dimensional analog of a polygon or
polyhedron. Chapter V of this book is entitled 'The Kaleidoscope'....)"

So what is Coxeter saying in relation to Derrida? I think this is
more the central issue. On the one hand images speak to what perception
is capable of, beyond normal eyesight and without concepts, reiterated in the nature of the discussion about animals. This is what
animals lack, given they do not have this conceptual ability, just that
they are able to deduct, was what I was looking for as that discussion
emerged and evolved.

If
there is a Platonic Ideal Form then there must be an ideal
representation of such a form. According to logocentrism, this ideal
representation is the logos.

Think of what the Good means again here that it cannot decay into
anything else when it is recognized, and that any other wording
degrades. If you can draw from experience then in a way one is able to
understand this. I had mention an archetype as a medium toward which one
could meet the good, and in that find that the archetype itself,
contain in the good, allows this insight to be shared. The whole scene
is the transmission of the idea, can become the ideal in life. This is
an immediate realization of the form of the good. It needs no further
clarification......at the deepest levels you recognize it. You know, and
you know it as a truth.

Understanding the foundations of Mathematics is important.

So I relay an instance where one is able to access the good.......also
in having mentioned that abstraction can lead to the good. This
distinction may have been settle in regard to the way in which Coxeter
sees and Derrida sees, in regards to the word, or how Coxeter sees
geometrically.

This is a crucial point in my view that such work could see the pattern
in the form of the good. This is as to say, and has been said, that such
freedom in realization is to know that the fifth postulate changed the
course of geometrical understandings. This set the future for how such
geometries would become significant in pushing not only Einstein
forward, but all that had followed him, by what Grossman learned of
Riemann. What Riemann learned from Gauss.

Sunday, December 21, 2014

I cannot say for certain and I speculate. Bucky balls then bring to mind
this architectural structure? Let me give you an example of a recent
discovery. I have to wonder if Bucky was a Platonist at heart......with grand ideas? Perhaps you recognze some Platonist idea about perfection as if mathematically a Tegmarkan might have found some truth? Some absolute truth? Perhaps a Penrose truth (Quasicrystal and Information)?

Aperiodic tilings serve as mathematical models for quasicrystals,
physical solids that were discovered in 1982 by Dan Shechtman[3] who
subsequently won the Nobel prize in 2011.[4] However, the specific local
structure of these materials is still poorly understood .Aperiodic tilings -

While one starts with a single point of entry......the whole process
from another perspective is encapsulated. So you might work from the
hydrogen spectrum as a start with the assumption, that this process in
itself is enclosed.

By now it is known theoretically that quantum angular momentum of any
kind has a discrete spectrum, which is sometimes imprecisely expressed
as "angular momentum is quantized".Stern–Gerlach experiment -

***

So possibly a Photon polarization principle inherent in
a quantum description of the wave and such a principle inherent in the
use of photosynthesis to describe a property not just of the capability
of using sun light, but of understanding this principle biologically in
human beings? I actually have a example of this use theoretically as a
product. Maybe Elon Musk might like to use it?

Photonic molecules are a synthetic form of matter in which photons bind together to form "molecules".
According to Mikhail Lukin, individual (massless) photons "interact
with each other so strongly that they act as though they have mass". The
effect is analogous to refraction. The light enters another medium,
transferring part of its energy to the medium. Inside the medium, it
exists as coupled light and matter, but it exits as light.[1]

While I would like to make it easy for you, I can only leave a title for
your examination. "The Nobel Prize in Physics 1914 Max von Laue." Yes, but if it is understood that some correlate process can be
understood from "a fundamental position," as to the architecture of
matter, what would this light have to say about the component
structuralism of the information we are missing?

The idea is not new. From a science fiction point of view, StarTrek had
these units that when you were hungry or wanted a drink you would have
this object materialize in a microwave type oven? Not the transporter.

So, you have this 3d printer accessing all information about the
structure and access to the building blocks of all matter in energy,
funneled through this replicator.

***

When Bucky was waving his arm between the earth and the moon.....did he
know about the three body problem, or how to look at the space between
these bodies in another way. If people think this is not real, then you
will have to tell those who use celestial mechanics that they are using
their satellite trajectories all wrong.

Ephemeralization, a term coined by R. Buckminster Fuller,
is the ability of technological advancement to do "more and more with
less and less until eventually you can do everything with nothing".[1]
Fuller's vision was that ephemeralization will result in
ever-increasing standards of living for an ever-growing population
despite finite resources.

Exactly. So it was not just "hand waving" Buckminister Fuller is
alluding too, but some actual understanding to "more is less?" One
applies the principle then? See? I am using your informational video to
explain.

ARTEMIS-P1 is the first spacecraft to navigate to and perform
stationkeeping operations around the Earth-Moon L1 and L2 Lagrangian
points. There are five Lagrangian points associated with the Earth-Moon
system. ARTEMIS - The First Earth-Moon Libration Orbiter -

To do more with less, it has to be understood that distance crossed
needs minimum usage of fuel to project the satellite over a great
distance. So they use "momentum" to swing satellites forward?

This is a list of various types of equilibrium, the condition of a system in which all competing influences are balanced. List of types of equilibrium -

Monday, June 16, 2014

"From future structural and kinematical studies of known quasicrystals, such as AlNiCo, these principles may be established providing a new understanding of and new control over the formation and structure of quasicrystals. See: A New Paradigm for the Structure of Quasicrystals

I really enjoyed the search for who supplied the original sample and from where. The journey back to the spot. Since following the subject of quasi-crystals for some time now, this journey was a nice addition to understanding the nature of matter in the early universe. This goes toward foundation, and my understanding of the work to piece together how nature sought to express itself materialistically from Reflection_symmetry as a representation of that early universe. I might have to be corrected here.

The concept of aperiodic crystal was coined by Erwin Schrödinger in another context with a somewhat different meaning. In his popular book What is life?
in 1944, Schrödinger sought to explain how hereditary information is
stored: molecules were deemed too small, amorphous solids were plainly
chaotic, so it had to be a kind of crystal; as a periodic structure
could not encode information, it had to be aperiodic. DNA
was later discovered, and, although not crystalline, it possesses
properties predicted by Schrödinger—it is a regular but aperiodic
molecule. See Also, with regard to Shrodinger:A New Physics Theory of Life

Friday, February 07, 2014

Unus mundus, Latin for "one world", is the concept of an underlying unified reality from which everything emerges and to which everything returns.The idea was popularized in the 20th century by the Swiss psychoanalyst Carl Jung, though the term can be traced back to scholastics such as Duns Scotus[1] and was taken up again in the 16th century by Gerhard Dorn, a student of the famous alchemist Paracelsus.

The striving for me was to dig deeper into our very natures. It always the quest to understand the patterns that reside in us. The very idea for me was that in this quest to unify, the objective world(matter) with the world that resides in a center place. To me that place was the source from which all things manifest.

Jung, in conjunction with the physicist Wolfgang Pauli, explored the possibility that his concepts of the archetype and synchronicity might be related to the unus mundus - the archetype being an expression of unus mundus; synchronicity, or "meaningful coincidence", being made possible by the fact that both the observer and connected phenomenon ultimately stem from the same source, the unus mundus.[2]

So while there was this objective striving to see how such formations emerged as materiality of such expression, was a final construct that existed in that external world. For me this was something no one could quite explain to me, yet, as I moved forward I began to find such correlates as to others who tried to map that expression.

It was this psychoid aspect of the archetype that so impressed Nobel laureate physicist Wolfgang Pauli. Embracing Jung's concept, Pauli believed that the archetype provided a link between physical events and the mind of the scientist who studied them. In doing so he echoed the position adopted by German astronomer Johannes Kepler.
Thus the archetypes which ordered our perceptions and ideas are
themselves the product of an objective order which transcends both the
human mind and the external world.[2]

This as the idea emerged, I looked for what emergence might mean, as an example of a beginning, and the subsequent model that may emerge from that source. This then became know as the "arche," and the tendency to form"(type)" as a movement forward in the solidifying of that expression. This was a matter bound expression, fully recognizing the need for a spiritual recognition of this opposition as a struggle in with consciousness to seek balance with materiality. Polarity, as the world of the real.

One of Duchamp's close friends Man Ray
(1890–1976) was also one of Duchamp's collaborators. His photograph
'Dust Breeding' (Duchamp's Large Glass with Dust Notes) from 1920 is a
document of The Large Glass after it had collected a year's worth of
dust while Duchamp was in New York. See: Dust Breeding (Man Ray 1920)

Such histrionically values were tied to such expressions to have found that the inner world and the outer-world were extremely connected. The observance not seen until it was understood that this psychology was topological interpreting itself from an inductive/deductive stance, as to the question, and with regard to the nature of the question.

Jung interpreted the practice of alchemy as the symbolic projection of psychic processes. In Psychology and Alchemy and Mysterium Coniunctionis (1955/56),
Jung’s empirical exploration and rediscovery of the objective psyche
led him to recognise that the basis of the alchemist’s endeavour was the
archetypal union of opposites by means of the integration of opposing
polarities: conscious and unconscious, reason and instinct, spiritual
and material, masculine and feminine. In the last summaries of his
insights on the subject, influenced in part by his collaboration with
the Nobel Prize winning physicist Wolfgang Pauli, the old Jung envisions
a great psycho-physical mystery to which the old alchemists gave the
name of unus mundus (one world). At the root of all being, so he
intimates, there is a state wherein physicality and spirituality meet.See:Reflections On Duchamp, Quantum Physics, And Mysterium Coniunctionis

This would place myself in the position of questioning this causal nature to have said that "will" was deeply connected to our psyche, to have not understood this deeper perception of a reality connection. Also, that such unification was deeper embedded in this practice of unification, so as to strive to form, as a example of an idea into expression.

This alchemy valuation of that work toward expression was based on a fundamental reality of joining the objectified world with the nature of the source. This forming process, the constructs, as a fundamental structure of the reality given.

Sunday, June 02, 2013

Albert Einstein Professor in Science, Departments of Physics and Astrophysical...

Quasi-elegance....As
a young student first reading Weyl's book, crystallography seemed like
the "ideal" of what one should be aiming for in science: elegant
mathematics that provides a complete understanding of all
physical possibilities. Ironically, many years later, I played a role
in showing that my "ideal" was seriously flawed. In 1984, Dan Shechtman,
Ilan Blech, Denis Gratias and John Cahn reported the discovery of a
puzzling manmade alloy of aluminumand manganese with icosahedral
symmetry. Icosahedral symmetry, with its six five-fold symmetry axes, is
the most famous forbidden crystal symmetry. As luck would have it, Dov
Levine (Technion) and I had been developing a hypothetical idea of a
new form of solid that we dubbed quasicrystals, short for quasiperiodic crystals. (A quasiperiodic
atomic arrangement means the atomic positions can be described by a
sum of oscillatory functions whose frequencies have an irrational
ratio.) We were inspired by a two-dimensional tiling invented by Sir
Roger Penrose known as the Penrose tiling, comprised of two tiles
arranged in a five-fold symmetric pattern. We showed that quasicrystals
could exist in three dimensions and were not subject to the rules of
crystallography. In fact, they could have any of the symmetries
forbidden to crystals. Furthermore, we showed that the diffraction
patterns predicted for icosahedral quasicrystals matched the Shechtman
et al. observations. Since 1984, quasicrystals with other forbidden
symmetries have been synthesized in the laboratory. The 2011 Nobel Prize
in Chemistry was awarded to Dan Shechtman for his experimental
breakthrough that changed our thinking about possible forms of matter.
More recently, colleagues and I have found evidence that quasicrystals
may have been among the first minerals to have formed in the solar
system.

The crystallography I first encountered in Weyl's book, thought to
be complete and immutable, turned out to be woefully incomplete,
missing literally an uncountable number of possible symmetries for
matter. Perhaps there is a lesson to be learned: While elegance and
simplicity are often useful criteria for judging theories, they can
sometimes mislead us into thinking we are right, when we are actually
infinitely wrong. See:2012 : WHAT IS YOUR FAVORITE DEEP, ELEGANT, OR BEAUTIFUL EXPLANATION?

"Clouds are not spheres, mountains are not cones, coastlines are not
circles, and bark is not smooth, nor does lightning travel in a straight
line." So writes acclaimed mathematician Benoit Mandelbrot in his
path-breaking book The Fractal Geometry of Nature. Instead,
such natural forms, and many man-made creations as well, are "rough,"
he says. To study and learn from such roughness, for which he invented
the term fractal, Mandelbrot devised a new kind of visual
mathematics based on such irregular shapes. Fractal geometry, as he
called this new math, is worlds apart from the Euclidean variety we all
learn in school, and it has sparked discoveries in myriad fields, from
finance to metallurgy, cosmology to medicine. In this interview, hear
from the father of fractals about why he disdains rules, why he
considers himself a philosopher, and why he abandons work on any given
advance in fractals as soon as it becomes popular. A Radical Mind

As I watch the dialogue between Bruce Lipton and Tom Campbell here, there were many things that helped my perspective understand the virtual world in relation to how the biology subject was presented. It is obvious then why Bruce Lipton likes the analogies Tom Campbell has to offer. The epiphanies Bruce is having along the road to his developing biological work is very important. It is how each time a person makes the leap that one must understand how individuals change, how societies change.

Okay so for one, the subject of fractals presents itself and the idea of process fractals and Geometry Fractals were presented in relation to each other. Now the talk moved onto the very thought of geometry presented in context sort of raised by ire even though I couldn't distinguish the differences. The virtual world analogy is still very unsettling to me.

So ya I have something to learn here.

I think my problem was with how such iteration may be schematically driven so as toidentify the pattern. Is to see this process reveal itself on a much larger scale. So when I looked at the Euclidean basis as a Newtonian expression the evolution toward relativity had to include the idea of Non Euclidean geometries. This was the natural evolution of the math that lies at the basis of graduating from a Euclidean world. It is the natural expression of understanding how this geometry can move into a dynamical world.

So yes the developing perspective for me is that even though we are talking abut mathematical structures here we see some correspondence in nature . This has been my thing so as to discover the starting point?

A schematic of a transmembrane receptor

It the truest sense I had already these questions in my mind as I was going through the talk. The starting point for Bruce is his biology and the cell. For Tom, he has not been explicit here other then to say that it is his studies with Monroe that he developed his thoughts around the virtual world as it relates to the idea of what he found working with Monroe.

So it is an exploration I feel of the work he encountered and has not so far as I seen made a public statement to that effect. It needs to be said and he needs to go back and look over how he had his epiphanies. For me this is about the process of discovery and creativity that I have found in my own life. Can one feel so full as to have found ones wealth in being that you can look everywhere and see the beginnings of many things?

This wealth is not monetary for me although I recognized we had to take care of or families and made sure they were ready to be off on their own. To be productive.

So for me the quest for that starting point is to identify the pattern that exists in nature as much as many have tried various perspective in terms of quantum gravity. Yes, we are all sort of like blind men trying to explain the reality of the world in our own way and in the process we may come up with our epiphanies.

These epiphanies help us to the next level of understanding as if we moved outside of our skeletal frame to allow the membrane of the cell to allow receptivity of what exist in the world around as information. We are not limited then to the frame of the skeleton hardened too, that we cannot progress further. The surface area of the membrane then becomes a request to open the channels toward expansion of the limitations we had applied to ourselves maintaining a frame of reference.

Monday, July 16, 2012

"The
end he (the artist) strives for is something else than a perfectly
executed print. His aim is to depict dreams, ideas, or problems in such a
way that other people can observe and consider them." - M.C. Escher

To them, I said,
the truth would be literally nothing
but the shadows of the images.
-Plato, The Republic (Book VII)

The idea here is about how one's observation and model perceptions
arises from some ordered perspective. Some use a starting point as an
assumption of position. Do recognize "the starting point" in the
previous examples?

In
one form or another, the issue of the ultimate beginning has engaged
philosophers and theologians in nearly every culture. It is entwined
with a grand set of concerns, one famously encapsulated in an 1897
painting by Paul Gauguin: D'ou venons-nous? Que sommes-nous? Ou
allons-nous? "Where do we come from? What are we? Where are we going?"

The effective realization that particle constructs are somehow
smaller windows of a much larger perspective fails to take in account
this idea that I am expressing as a foundational approach to that
starting point.

If you do not go all the way toward defining of that "point of
equilibrium" how are you to understand how information is easily
transferred to the individual from a much larger reality of existence?
One would assume information is all around us? That there are multitudes
of pathways that allow us to arrive at some some probability density
configuration as some measure of an Pascalian ideal.

Of course there are problems with this in terms of our defining a heat death in individuals?

That's not possible so one is missing the understanding here about
equilibrium. I might have said we are positional in terms of the past
and the future with regard to memory and the anticipated future? How is
that heat death correlated? It can't.

So you have to look for examples in relation to how one may arrive at
that beginning point. Your theory may not be sufficiently dealing with
the information as it is expressed in terms of your approach to the
small window?

There are mathematical inspections here that have yet to be associated
with more then discrete functions of reality as expressive building
blocks of interpretation. The basic assumption of discrete function
still exists in contrast to continuity of expression. This is the
defining realization in assuming the model that MBT provides. I have meet
the same logic in the differences of scientific approach toward the
definition of what is becoming?

On the one hand, a configuration space as demonstrated by Tom that is
vastly used in science. On the other, a recognition of how thick in
measure viscosity is realized and what the physics is in this
association. Not just the physical manifestation of, but of what happens
when equilibrium is reached. Hot or very cold. Temperature, is not a problem then?

See my problem is that I can show you levitation of objects using
superconductors but I cannot produce this in real life without that
science. Yet, in face of that science I know that something can happen
irregardless of what all the science said, so I am looking as well to
combining the meta with the physical to realize that such a conditions
may arise in how we as a total culture have accepted the parameters of
our thinking.

So by dealing with those parameters I too hoped to see a cultural
shift(paradigm and Kuhn) by adoption of the realization as we are with
regard to the way in which we function in this reality. So if your
thinking abut gravity how is this possible within the "frame work" to
have it encroach upon our very own psychological makeup too?

Friday, June 29, 2012

This image depicts the interaction of nine plane waves—expanding sets
of ripples, like the waves you would see if you simultaneously dropped
nine stones into a still pond. The pattern is called a quasicrystal
because it has an ordered structure, but the structure never repeats
exactly. The waves produced by dropping four or more stones into a pond
always form a quasicrystal.

Because of the wavelike properties of matter at subatomic scales,
this pattern could also be seen in the waveform that describes the
location of an electron. Harvard physicist Eric Heller created this
computer rendering and added color to make the pattern’s structure
easier to see. See:What Is This?
A Psychedelic Place Mat?

In 1941, Escher wrote his first paper, now publicly recognized, called Regular Division of the Plane with Asymmetric Congruent Polygons,
which detailed his mathematical approach to artwork creation. His
intention in writing this was to aid himself in integrating mathematics
into art. Escher is considered a research mathematician of his time
because of his documentation with this paper. In it, he studied color
based division, and developed a system of categorizing combinations of
shape, color and symmetrical properties. By studying these areas, he
explored an area that later mathematicians labeled crystallography.
Around 1956, Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher's works Circle Limit I–IV
demonstrate this concept. In 1995, Coxeter verified that Escher had
achieved mathematical perfection in his etchings in a published paper.
Coxeter wrote, "Escher got it absolutely right to the millimeter."

Snow Crystal Photo Gallery I

If you have never studied the structure of Mandala origins of the Tibetan Buddhist you might never of recognize the structure given to this 2 dimensional surface? Rotate the 2d surface to the side view. It becomes a recognition of some Persian temple perhaps? I mean, the video really helps one to see this, and to understand the structural integrity is built upon.

So too, do we recognize this "snow flake" as some symmetrical realization of it's individuality as some mathematical form constructed in nature?

I previous post I gave some inclination to the idea of time travel and how this is done within the scope of consciousness. In the same vein, I want you to realize that such journeys to our actualized past can bring us in contact with a book of Mandalas that helped me to realize and reveals a key of symmetrical expressions of the lifetime, or lifetimes.

Again in relation how science sees subjectivity I see that this is weak in expression in terms of how it can be useful in an objective sense as to be repeatable. But it helps too, to trace this beginning back to a source that while perceived as mathematical , shows the the mathematical relation embedded in nature.

Thursday, January 05, 2012

In mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystallineliquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry. Patterns are located upon the points of a lattice, which is an array of points repeating periodically in three dimensions. The points can be thought of as forming identical tiny boxes, called unit cells, that fill the space of the lattice. The lengths of the edges of a unit cell and the angles between them are called the lattice parameters. The symmetry properties of the crystal are embodied in its space group.

Unit cell

The crystal structure of a material or the arrangement of atoms within a given type of crystal structure can be described in terms of its unit cell. The unit cell is a small box containing one or more atoms, a spatial arrangement of atoms. The unit cells stacked in three-dimensional space describe the bulk arrangement of atoms of the crystal. The crystal structure has a three-dimensional shape. The unit cell is given by its lattice parameters, which are the length of the cell edges and the angles between them, while the positions of the atoms inside the unit cell are described by the set of atomic positions (xi , yi , zi) measured from a lattice point.

Simple cubic (P)

Body-centered cubic (I)

Face-centered cubic (F)

Miller indices

Vectors and atomic planes in a crystal lattice can be described by a three-value Miller index notation (ℓmn). The ℓ, m, and n directional indices are separated by 90°, and are thus orthogonal. In fact, the ℓ component is mutually perpendicular to the m and n indices.

By definition, (ℓmn) denotes a plane that intercepts the three points a1/ℓ, a2/m, and a3/n, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell (in the basis of the lattice vectors). If one or more of the indices is zero, it simply means that the planes do not intersect that axis (i.e., the intercept is "at infinity").

Considering only (ℓmn) planes intersecting one or more lattice points (the lattice planes), the perpendicular distance d between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula:

Planes and directions

The crystallographic directions are fictitious lines linking nodes (atoms, ions or molecules) of a crystal. Likewise, the crystallographic planes are fictitious planes linking nodes. Some directions and planes have a higher density of nodes. These high density planes have an influence on the behavior of the crystal as follows:

Adsorption and reactivity: Physical adsorption and chemical reactions occur at or near surface atoms or molecules. These phenomena are thus sensitive to the density of nodes.

Surface tension: The condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species. The surface tension of an interface thus varies according to the density on the surface.

Plastic deformation: Dislocation glide occurs preferentially parallel to higher density planes. The perturbation carried by the dislocation (Burgers vector) is along a dense direction. The shift of one node in a more dense direction requires a lesser distortion of the crystal lattice.

In the rhombohedral, hexagonal, and tetragonal systems, the basal plane is the plane perpendicular to the principal axis.

Cubic structures

For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted a); similarly for the reciprocal lattice. So, in this common case, the Miller indices (ℓmn) and [ℓmn] both simply denote normals/directions in Cartesian coordinates. For cubic crystals with lattice constanta, the spacing d between adjacent (ℓmn) lattice planes is (from above):

Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes:

Coordinates in angle brackets such as <100> denote a family of directions that are equivalent due to symmetry operations, such as [100], [010], [001] or the negative of any of those directions.

Coordinates in curly brackets or braces such as {100} denote a family of plane normals that are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.

Classification

The defining property of a crystal is its inherent symmetry, by which we mean that under certain 'operations' the crystal remains unchanged. For example, rotating the crystal 180° about a certain axis may result in an atomic configuration that is identical to the original configuration. The crystal is then said to have a twofold rotational symmetry about this axis. In addition to rotational symmetries like this, a crystal may have symmetries in the form of mirror planes and translational symmetries, and also the so-called "compound symmetries," which are a combination of translation and rotation/mirror symmetries. A full classification of a crystal is achieved when all of these inherent symmetries of the crystal are identified.[1]

Lattice systems

These lattice systems are a grouping of crystal structures according to the axial system used to describe their lattice. Each lattice system consists of a set of three axes in a particular geometrical arrangement. There are seven lattice systems. They are similar to but not quite the same as the seven crystal systems and the six crystal families.

Atomic coordination

By considering the arrangement of atoms relative to each other, their coordination numbers (or number of nearest neighbors), interatomic distances, types of bonding, etc., it is possible to form a general view of the structures and alternative ways of visualizing then.

HCP lattice (left) and the fcc lattice (right).

Close packing

The principles involved can be understood by considering the most efficient way of packing together equal-sized spheres and stacking close-packed atomic planes in three dimensions. For example, if plane A lies beneath plane B, there are two possible ways of placing an additional atom on top of layer B. If an additional layer was placed directly over plane A, this would give rise to the following series :

...ABABABAB....

This type of crystal structure is known as hexagonal close packing (hcp).

If however, all three planes are staggered relative to each other and it is not until the fourth layer is positioned directly over plane A that the sequence is repeated, then the following sequence arises:

...ABCABCABC...

This type of crystal structure is known as cubic close packing (ccp)

The unit cell of the ccp arrangement is the face-centered cubic (fcc) unit cell. This is not immediately obvious as the closely packed layers are parallel to the {111} planes of the fcc unit cell. There are four different orientations of the close-packed layers.

The packing efficiency could be worked out by calculating the total volume of the spheres and dividing that by the volume of the cell as follows:

The 74% packing efficiency is the maximum density possible in unit cells constructed of spheres of only one size. Most crystalline forms of metallic elements are hcp, fcc, or bcc (body-centered cubic). The coordination number of hcp and fcc is 12 and its atomic packing factor (APF) is the number mentioned above, 0.74. The APF of bcc is 0.68 for comparison.

Bravais lattices

When the crystal systems are combined with the various possible lattice centerings, we arrive at the Bravais lattices. They describe the geometric arrangement of the lattice points, and thereby the translational symmetry of the crystal. In three dimensions, there are 14 unique Bravais lattices that are distinct from one another in the translational symmetry they contain. All crystalline materials recognized until now (not including quasicrystals) fit in one of these arrangements. The fourteen three-dimensional lattices, classified by crystal system, are shown above. The Bravais lattices are sometimes referred to as space lattices.

The crystal structure consists of the same group of atoms, the basis, positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the 14 Bravais lattices. The characteristic rotation and mirror symmetries of the group of atoms, or unit cell, is described by its crystallographic point group.

Point groups

The crystallographic point group or crystal class is the mathematical group comprising the symmetry operations that leave at least one point unmoved and that leave the appearance of the crystal structure unchanged. These symmetry operations include

Reflection, which reflects the structure across a reflection plane

Rotation, which rotates the structure a specified portion of a circle about a rotation axis

Inversion, which changes the sign of the coordinate of each point with respect to a center of symmetry or inversion point

Improper rotation, which consists of a rotation about an axis followed by an inversion.

Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively called symmetry elements. There are 32 possible crystal classes. Each one can be classified into one of the seven crystal systems.

Space groups

The space group of the crystal structure is composed of the translational symmetry operations in addition to the operations of the point group. These include:

Pure translations, which move a point along a vector

Screw axes, which rotate a point around an axis while translating parallel to the axis

Glide planes, which reflect a point through a plane while translating it parallel to the plane.

There are 230 distinct space groups.

Grain boundaries

Grain boundaries are interfaces where crystals of different orientations meet. A grain boundary is a single-phase interface, with crystals on each side of the boundary being identical except in orientation. The term "crystallite boundary" is sometimes, though rarely, used. Grain boundary areas contain those atoms that have been perturbed from their original lattice sites, dislocations, and impurities that have migrated to the lower energy grain boundary.

Treating a grain boundary geometrically as an interface of a single crystal cut into two parts, one of which is rotated, we see that there are five variables required to define a grain boundary. The first two numbers come from the unit vector that specifies a rotation axis. The third number designates the angle of rotation of the grain. The final two numbers specify the plane of the grain boundary (or a unit vector that is normal to this plane).

Grain boundaries disrupt the motion of dislocations through a material, so reducing crystallite size is a common way to improve strength, as described by the Hall–Petch relationship. Since grain boundaries are defects in the crystal structure they tend to decrease the electrical and thermal conductivity of the material. The high interfacial energy and relatively weak bonding in most grain boundaries often makes them preferred sites for the onset of corrosion and for the precipitation of new phases from the solid. They are also important to many of the mechanisms of creep.

Grain boundaries are in general only a few nanometers wide. In common materials, crystallites are large enough that grain boundaries account for a small fraction of the material. However, very small grain sizes are achievable. In nanocrystalline solids, grain boundaries become a significant volume fraction of the material, with profound effects on such properties as diffusion and plasticity. In the limit of small crystallites, as the volume fraction of grain boundaries approaches 100%, the material ceases to have any crystalline character, and thus becomes an amorphous solid.

Defects and impurities

Real crystals feature defects or irregularities in the ideal arrangements described above and it is these defects that critically determine many of the electrical and mechanical properties of real materials. When one atom substitutes for one of the principal atomic components within the crystal structure, alteration in the electrical and thermal properties of the material may ensue.[2] Impurities may also manifest as spin impurities in certain materials. Research on magnetic impurities demonstrates that substantial alteration of certain properties such as specific heat may be affected by small concentrations of an impurity, as for example impurities in semiconducting ferromagneticalloys may lead to different properties as first predicted in the late 1960s.[3][4]Dislocations in the crystal lattice allow shear at lower stress than that needed for a perfect crystal structure.[5]

Prediction of structure

The difficulty of predicting stable crystal structures based on the knowledge of only the chemical composition has long been a stumbling block on the way to fully computational materials design. Now, with more powerful algorithms and high-performance computing, structures of medium complexity can be predicted using such approaches as evolutionary algorithms, random sampling, or metadynamics.

The crystal structures of simple ionic solids (e.g., NaCl or table salt) have long been rationalized in terms of Pauling's rules, first set out in 1929 by Linus Pauling, referred to by many since as the "father of the chemical bond".[6] Pauling also considered the nature of the interatomic forces in metals, and concluded that about half of the five d-orbitals in the transition metals are involved in bonding, with the remaining nonbonding d-orbitals being responsible for the magnetic properties. He, therefore, was able to correlate the number of d-orbitals in bond formation with the bond length as well as many of the physical properties of the substance. He subsequently introduced the metallic orbital, an extra orbital necessary to permit uninhibited resonance of valence bonds among various electronic structures.[7]

In the resonating valence bond theory, the factors that determine the choice of one from among alternative crystal structures of a metal or intermetallic compound revolve around the energy of resonance of bonds among interatomic positions. It is clear that some modes of resonance would make larger contributions (be more mechanically stable than others), and that in particular a simple ratio of number of bonds to number of positions would be exceptional. The resulting principle is that a special stability is associated with the simplest ratios or "bond numbers": 1/2, 1/3, 2/3, 1/4, 3/4, etc. The choice of structure and the value of the axial ratio (which determines the relative bond lengths) are thus a result of the effort of an atom to use its valency in the formation of stable bonds with simple fractional bond numbers.[8][9]

After postulating a direct correlation between electron concentration and crystal structure in beta-phase alloys, Hume-Rothery analyzed the trends in melting points, compressibilities and bond lengths as a function of group number in the periodic table in order to establish a system of valencies of the transition elements in the metallic state. This treatment thus emphasized the increasing bond strength as a function of group number.[10] The operation of directional forces were emphasized in one article on the relation between bond hybrids and the metallic structures. The resulting correlation between electronic and crystalline structures is summarized by a single parameter, the weight of the d-electrons per hybridized metallic orbital. The “d-weight” calculates out to 0.5, 0.7 and 0.9 for the fcc, hcp and bcc structures respectively. The relationship between d-electrons and crystal structure thus becomes apparent.[11]

Polymorphism refers to the ability of a solid to exist in more than one crystalline form or structure. According to Gibbs' rules of phase equilibria, these unique crystalline phases will be dependent on intensive variables such as pressure and temperature. Polymorphism can potentially be found in many crystalline materials including polymers, minerals, and metals, and is related to allotropy, which refers to elemental solids. The complete morphology of a material is described by polymorphism and other variables such as crystal habit, amorphous fraction or crystallographic defects. Polymorphs have different stabilities and may spontaneously convert from a metastable form (or thermodynamically unstable form) to the stable form at a particular temperature. They also exhibit different melting points, solubilities, and X-ray diffraction patterns.

One good example of this is the quartz form of silicon dioxide, or SiO2. In the vast majority of silicates, the Si atom shows tetrahedral coordination by 4 oxygens. All but one of the crystalline forms involve tetrahedral SiO4 units linked together by shared vertices in different arrangements. In different minerals the tetrahedra show different degrees of networking and polymerization. For example, they occur singly, joined together in pairs, in larger finite clusters including rings, in chains, double chains, sheets, and three-dimensional frameworks. The minerals are classified into groups based on these structures. In each of its 7 thermodynamically stable crystalline forms or polymorphs of crystalline quartz, only 2 out of 4 of each the edges of the SiO4 tetrahedra are shared with others, yielding the net chemical formula for silica: SiO2.
Another example is elemental tin (Sn), which is malleable near ambient temperatures but is brittle when cooled. This change in mechanical properties due to existence of its two major allotropes, α- and β-tin. The two allotropes that are encountered at normal pressure and temperature, α-tin and β-tin, are more commonly known as gray tin and white tin respectively. Two more allotropes, γ and σ, exist at temperatures above 161 °C and pressures above several GPa.[12] White tin is metallic, and is the stable crystalline form at or above room temperature. Below 13.2 °C, tin exists in the gray form, which has a diamond cubic crystal structure, similar to diamond, silicon or germanium. Gray tin has no metallic properties at all, is a dull-gray powdery material, and has few uses, other than a few specialized semiconductor applications.[13] Although the α-β transformation temperature of tin is nominally 13.2 °C, impurities (e.g. Al, Zn, etc.) lower the transition temperature well below 0 °C, and upon addition of Sb or Bi the transformation may not occur at all.[14]

Physical properties

Twenty of the 32 crystal classes are so-called piezoelectric, and crystals belonging to one of these classes (point groups) display piezoelectricity. All piezoelectric classes lack a centre of symmetry. Any material develops a dielectric polarization when an electric field is applied, but a substance that has such a natural charge separation even in the absence of a field is called a polar material. Whether or not a material is polar is determined solely by its crystal structure. Only 10 of the 32 point groups are polar. All polar crystals are pyroelectric, so the 10 polar crystal classes are sometimes referred to as the pyroelectric classes.

There are a few crystal structures, notably the perovskite structure, which exhibit ferroelectric behavior. This is analogous to ferromagnetism, in that, in the absence of an electric field during production, the ferroelectric crystal does not exhibit a polarization. Upon the application of an electric field of sufficient magnitude, the crystal becomes permanently polarized. This polarization can be reversed by a sufficiently large counter-charge, in the same way that a ferromagnet can be reversed. However, it is important to note that, although they are called ferroelectrics, the effect is due to the crystal structure (not the presence of a ferrous metal).